The circle method was developed by Hardy and Littlewood in order to handle additive problems in the number theory. It was significantly improved by Vinogradov [1]} in 1928 and was used by him to obtain the asymptotic formula in the Goldbach ternary representation problem. The main innovation introduced by Vinogradov was the usage of the exponential sums instead of infinite series which greatly simplified the method. Usually the application of the circle method is followed by using estimates for the exponential sums like Weyl's or Vinogradov's inequality. Over the years the circle method has become a widely used tool in analytic number theory as well as in harmonic analysis, see [2]}, [3]}, [4]}, [5]} and references given there. We recommend an excellent treatise on the subject due to Vaughan [6]}. For more general view on the analytic number theory we refer to the monograph by Iwaniec and Kowalski [7]}.
